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<div class="titlepage"><div><div><h3 class="title">
<a name="math_toolkit.roots_noderiv.bracket_solve"></a><a class="link" href="bracket_solve.html" title="Bracket and Solve Root">Bracket and
      Solve Root</a>
</h3></div></div></div>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Tol</span><span class="special">&gt;</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">&gt;</span>
   <span class="identifier">bracket_and_solve_root</span><span class="special">(</span>
      <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
      <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">guess</span><span class="special">,</span>
      <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">factor</span><span class="special">,</span>
      <span class="keyword">bool</span> <span class="identifier">rising</span><span class="special">,</span>
      <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
      <span class="identifier">std</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&amp;</span> <span class="identifier">max_iter</span><span class="special">);</span>

<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Tol</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">&gt;</span>
   <span class="identifier">bracket_and_solve_root</span><span class="special">(</span>
      <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
      <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">guess</span><span class="special">,</span>
      <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&amp;</span> <span class="identifier">factor</span><span class="special">,</span>
      <span class="keyword">bool</span> <span class="identifier">rising</span><span class="special">,</span>
      <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
      <span class="identifier">std</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&amp;</span> <span class="identifier">max_iter</span><span class="special">,</span>
      <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
</pre>
<p>
        <code class="computeroutput"><span class="identifier">bracket_and_solve_root</span></code> is
        a convenience function that calls <a class="link" href="TOMS748.html" title="Algorithm TOMS 748: Alefeld, Potra and Shi: Enclosing zeros of continuous functions">TOMS
        748 algorithm</a> internally to find the root of <span class="emphasis"><em>f(x)</em></span>.
        It is generally much easier to use this function rather than <a class="link" href="TOMS748.html" title="Algorithm TOMS 748: Alefeld, Potra and Shi: Enclosing zeros of continuous functions">TOMS
        748 algorithm</a>, since it does the hard work of bracketing the root
        for you. It's bracketing routines are quite robust and will usually be more
        foolproof than home-grown routines, unless the function can be analysed to
        yield tight brackets.
      </p>
<p>
        Note that this routine can only be used when:
      </p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
            <span class="emphasis"><em>f(x)</em></span> is monotonic in the half of the real axis containing
            <span class="emphasis"><em>guess</em></span>.
          </li>
<li class="listitem">
            The value of the initial guess must have the same sign as the root: the
            function will <span class="emphasis"><em>never cross the origin</em></span> when searching
            for the root.
          </li>
<li class="listitem">
            The location of the root should be known at least approximately, if the
            location of the root differs by many orders of magnitude from <span class="emphasis"><em>guess</em></span>
            then many iterations will be needed to bracket the root in spite of the
            special heuristics used to guard against this very situation. A typical
            example would be setting the initial guess to 0.1, when the root is at
            1e-300.
          </li>
</ul></div>
<p>
        The <code class="computeroutput"><span class="identifier">bracket_and_solve_root</span></code>
        parameters are:
      </p>
<div class="variablelist">
<p class="title"><b></b></p>
<dl class="variablelist">
<dt><span class="term">f</span></dt>
<dd><p>
              A unary functor (or C++ lambda) that is the function whose root is
              to be solved. <span class="emphasis"><em>f(x)</em></span> must be uniformly increasing
              or decreasing on <span class="emphasis"><em>x</em></span>.
            </p></dd>
<dt><span class="term">guess</span></dt>
<dd><p>
              An initial approximation to the root.
            </p></dd>
<dt><span class="term">factor</span></dt>
<dd><p>
              A scaling factor that is used to bracket the root: the value <span class="emphasis"><em>guess</em></span>
              is multiplied (or divided as appropriate) by <span class="emphasis"><em>factor</em></span>
              until two values are found that bracket the root. A value such as 2
              is a typical choice for <span class="emphasis"><em>factor</em></span>. In addition <span class="emphasis"><em>factor</em></span>
              will be multiplied by 2 every 32 iterations: this is to guard against
              a really very bad initial guess, typically these occur when it's known
              the result is very large or small, but not the exact order of magnitude.
            </p></dd>
<dt><span class="term">rising</span></dt>
<dd><p>
              Set to <span class="emphasis"><em>true</em></span> if <span class="emphasis"><em>f(x)</em></span> is rising
              on <span class="emphasis"><em>x</em></span> and <span class="emphasis"><em>false</em></span> if <span class="emphasis"><em>f(x)</em></span>
              is falling on <span class="emphasis"><em>x</em></span>. This value is used along with
              the result of <span class="emphasis"><em>f(guess)</em></span> to determine if <span class="emphasis"><em>guess</em></span>
              is above or below the root.
            </p></dd>
<dt><span class="term">tol</span></dt>
<dd><p>
              A binary functor (or C++ lambda) that determines the termination condition
              for the search for the root. <span class="emphasis"><em>tol</em></span> is passed the
              current brackets at each step, when it returns true then the current
              brackets are returned as the pair result. See also <a class="link" href="root_termination.html" title="Termination Condition Functors">predefined
              termination functors</a>.
            </p></dd>
<dt><span class="term">max_iter</span></dt>
<dd><p>
              The maximum number of function invocations to perform in the search
              for the root. On exit is set to the actual number of invocations performed.
            </p></dd>
</dl>
</div>
<p>
        The final <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
        be used to control the behaviour of the function: how it handles errors,
        what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">policy
        documentation for more details</a>.
      </p>
<p>
        <span class="bold"><strong>Returns</strong></span>: a pair of values <span class="emphasis"><em>r</em></span>
        that bracket the root so that:
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          f(r.first) * f(r.second) &lt;= 0
        </p></blockquote></div>
<p>
        and either
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          tol(r.first, r.second) == true
        </p></blockquote></div>
<p>
        or
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          max_iter &gt;= m
        </p></blockquote></div>
<p>
        where <span class="emphasis"><em>m</em></span> is the initial value of <span class="emphasis"><em>max_iter</em></span>
        passed to the function.
      </p>
<p>
        In other words, it's up to the caller to verify whether termination occurred
        as a result of exceeding <span class="emphasis"><em>max_iter</em></span> function invocations
        (easily done by checking the value of <span class="emphasis"><em>max_iter</em></span> when
        the function returns), rather than because the termination condition <span class="emphasis"><em>tol</em></span>
        was satisfied.
      </p>
</div>
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<td align="right"><div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
      Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
      Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
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      Walker and Xiaogang Zhang<p>
        Distributed under the Boost Software License, Version 1.0. (See accompanying
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